| L(s) = 1 | + 1.28i·2-s − 79.7i·3-s + 126.·4-s + 102.·6-s − 538. i·7-s + 326. i·8-s − 4.17e3·9-s − 1.21e3·11-s − 1.00e4i·12-s − 7.07e3i·13-s + 690.·14-s + 1.57e4·16-s − 3.34e3i·17-s − 5.34e3i·18-s − 2.21e4·19-s + ⋯ |
| L(s) = 1 | + 0.113i·2-s − 1.70i·3-s + 0.987·4-s + 0.193·6-s − 0.593i·7-s + 0.225i·8-s − 1.90·9-s − 0.275·11-s − 1.68i·12-s − 0.892i·13-s + 0.0672·14-s + 0.961·16-s − 0.165i·17-s − 0.216i·18-s − 0.741·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.976275 - 1.57964i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.976275 - 1.57964i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 1.28iT - 128T^{2} \) |
| 3 | \( 1 + 79.7iT - 2.18e3T^{2} \) |
| 7 | \( 1 + 538. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 1.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.07e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 3.34e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 2.21e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.85e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 2.06e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.77e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.84e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.27e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.64e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 4.49e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 7.30e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.42e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.66e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.95e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 9.21e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.25e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 6.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.17e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 2.42e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.59e6iT - 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.60796488555370860054400663286, −14.10307226955372543024812885412, −12.94524770710582769492648774755, −11.93796822209457932641513948213, −10.62865125585346043551188752088, −8.082633958834805946813752869178, −7.18267693896448018691455166348, −6.00164948788106819785934418750, −2.61790461626448249160965604897, −0.997140075851543786042063485262,
2.71455852186026632147761716915, 4.50645350270869313840225724863, 6.24537318745991977135195371977, 8.573498343664183540434053416736, 10.00704344742970645793361027052, 10.96981394272647211630745315801, 12.14527952483116073483707141739, 14.40567765052595161883882340086, 15.43588668335975055411006839319, 16.11384204146282113152817966207
